A New Identity for the Least-square Solution of Overdetermined Set of Linear Equations

In this paper, we prove a new identity for the least-square solution of an over-determined set of linear equation Ax = b, where A is an m×n full-rank matrix, b is a column-vector of dimension m, and m (the number of equations) is larger than or equal to n (the dimension of the unknown vector x). Generally, the equations are inconsistent and there is no feasible solution for x unless b belongs to the column-span of A. In the least-square approach, a candidate solution is found as the unique x that minimizes the error function ‖Ax− b‖2. We propose a more general approach that consist in considering all the consistent subset of the equations, finding their solutions, and taking a weighted average of them to build a candidate solution. In particular, we show that by weighting the solutions with the squared determinant of their coefficient matrix, the resulting candidate solution coincides with the least square solution. Index Terms Over-determined linear equation, Least square solution.